how do you prove the reduction identities using a unit circle diagram?

Question

Reduction Identities in trigonometry such as:
Sin(x+pi)= -sin(x)
Cos(x+pi)= -cos(x).

Answer 1

See the picture:

http://mudandmuck.com/str2/sin&cos.jpg

The two triangles in the picture are equivalent. (Same angles, and one side of each we know to be the same (hypotenuse=1), so all the sides have the same absolute length, only their signs are different.)

Answer 2

by saying that the y coordinate on the unit circle is the sin x, and the x coord is the cos x

Answer 3

Let P be a point on the unit circle coordiantes (cosθ, sinθ)
where x = angle between positive x axis and the radius OP

Well by Pythagoras' rule
x² + y² = 1
So cos²θ + sin²θ = 1
tanθ = y/x = sinθ/cosθ

A rotation of π/2 counterclockwise converts (x, y) to (-y, x)
Thus sin(θ + π/2) = -cosθ and cos(θ + π/2) = sinθ

Similarly a rotation of π/2 clockwise converts (x, y) to (y, -x)
So sin(θ - π/2) = cosθ and cos(θ - π/2) = -sinθ

A rotation of π counterclockwise is the same as a rotation of π clockwise and

So sin(θ ± π) = -sinθ and cos(θ ± π) = -cosθ

A rotation of θ from the y- axis converts (x, y) to (y, x)
So sin(π/2 - θ) = cosθ and cos(π/2 - θ) = sinθ

A rotation of 2π counterclockwise and clockwise returns the point (x. y) to itself
So sin(θ + 2π) = sinθ and cos(θ + 2π) = cosθ

A reflection in the x-axis converts (x, y) to (x, -y)
So cos(-θ) = cosθ and sin(-θ) = -sinθ

A reflection in the y-axis converts (x, y) to (-x, y)
So cos(π - θ) = -cosθ and sin(π - θ) = sinθ

How many more do you need??